Quasi-polynomial Hitting-set for Set-depth-Delta Formulas

We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition (X_1,...,X_d) of the variable indices [n] that the top product layer respects, i.e. C(x) = \sum_{i=1}^k \prod_{j=1}^{d} f_{i,j}(x_{X_j}), where f_{i,j} is a sparse polynomial in F[x_{X_j}]. Extending this definition to any depth - we call a depth-Delta formula C (consisting of alternating layers of Sigma and Pi gates, with a Sigma-gate on top) a set-depth-Delta formula if every Pi-layer in C respects a (unknown) partition on the variables; if Delta is even then the product gates of the bottom-most Pi-layer are allowed to compute arbitrary monomials. In this work, we give a hitting-set generator for set-depth-Delta formulas (over any field) with running time polynomial in exp(({Delta}^2 log s)^{Delta - 1}), where s is the size bound on the input set-depth-Delta formula. In other words, we give a quasi-polynomial time blackbox polynomial identity test for such constant-depth formulas. Previously, the very special case of Delta=3 (also known as set-multilinear depth-3 circuits) had no known sub-exponential time hitting-set generator. This was declared as an open problem by Shpilka & Yehudayoff (FnT-TCS 2010); the model being first studied by Nisan & Wigderson (FOCS 1995). Our work settles this question, not only for depth-3 but, up to depth epsilon.log s / loglog s, for a fixed constant epsilon < 1. The technique is to investigate depth-Delta formulas via depth-(Delta-1) formulas over a Hadamard algebra, after applying a `shift' on the variables. We propose a new algebraic conjecture about the low-support rank-concentration in the latter formulas, and manage to prove it in the case of set-depth-Delta formulas.Comment: 22 page

Independence in Algebraic Complexity Theory

This thesis examines the concepts of linear and algebraic independence in algebraic complexity theory. Arithmetic circuits, computing multivariate polynomials over a field, form the framework of our complexity considerations. We are concerned with polynomial identity testing (PIT), the problem of deciding whether a given arithmetic circuit computes the zero polynomial. There are efficient randomized algorithms known for this problem, but as yet deterministic polynomial-time algorithms could be found only for restricted circuit classes. We are especially interested in blackbox algorithms, which do not inspect the given circuit, but solely evaluate it at some points. Known approaches to the PIT problem are based on the notions of linear independence and rank of vector subspaces of the polynomial ring. We generalize those methods to algebraic independence and transcendence degree of subalgebras of the polynomial ring. Thereby, we obtain efficient blackbox PIT algorithms for new circuit classes. The Jacobian criterion constitutes an efficient characterization for algebraic independence of polynomials. However, this criterion is valid only in characteristic zero. We deduce a novel Jacobian-like criterion for algebraic independence of polynomials over finite fields. We apply it to obtain another blackbox PIT algorithm and to improve the complexity of testing the algebraic independence of arithmetic circuits over finite fields.Die vorliegende Arbeit untersucht die Konzepte der linearen und algebraischen Unabhängigkeit innerhalb der algebraischen Komplexitätstheorie. Arithmetische Schaltkreise, die multivariate Polynome über einem Körper berechnen, bilden die Grundlage unserer Komplexitätsbetrachtungen. Wir befassen uns mit dem polynomial identity testing (PIT) Problem, bei dem entschieden werden soll ob ein gegebener Schaltkreis das Nullpolynom berechnet. Für dieses Problem sind effiziente randomisierte Algorithmen bekannt, aber deterministische Polynomialzeitalgorithmen konnten bisher nur für eingeschränkte Klassen von Schaltkreisen angegeben werden. Besonders von Interesse sind Blackbox-Algorithmen, welche den gegebenen Schaltkreis nicht inspizieren, sondern lediglich an Punkten auswerten. Bekannte Ansätze für das PIT Problem basieren auf den Begriffen der linearen Unabhängigkeit und des Rangs von Untervektorräumen des Polynomrings. Wir übertragen diese Methoden auf algebraische Unabhängigkeit und den Transzendenzgrad von Unteralgebren des Polynomrings. Dadurch erhalten wir effiziente Blackbox-PIT-Algorithmen für neue Klassen von Schaltkreisen. Eine effiziente Charakterisierung der algebraischen Unabhängigkeit von Polynomen ist durch das Jacobi-Kriterium gegeben. Dieses Kriterium ist jedoch nur in Charakteristik Null gültig. Wir leiten ein neues Jacobi-artiges Kriterium für die algebraische Unabhängigkeit von Polynomen über endlichen Körpern her. Dieses liefert einen weiteren Blackbox-PIT-Algorithmus und verbessert die Komplexität des Problems arithmetische Schaltkreise über endlichen Körpern auf algebraische Unabhängigkeit zu testen

Polynomial-Time Pseudodeterministic Construction of Primes

A randomized algorithm for a search problem is *pseudodeterministic* if it produces a fixed canonical solution to the search problem with high probability. In their seminal work on the topic, Gat and Goldwasser posed as their main open problem whether prime numbers can be pseudodeterministically constructed in polynomial time. We provide a positive solution to this question in the infinitely-often regime. In more detail, we give an *unconditional* polynomial-time randomized algorithm $B$ such that, for infinitely many values of $n$, $B(1^n)$ outputs a canonical $n$-bit prime $p_n$ with high probability. More generally, we prove that for every dense property $Q$ of strings that can be decided in polynomial time, there is an infinitely-often pseudodeterministic polynomial-time construction of strings satisfying $Q$. This improves upon a subexponential-time construction of Oliveira and Santhanam. Our construction uses several new ideas, including a novel bootstrapping technique for pseudodeterministic constructions, and a quantitative optimization of the uniform hardness-randomness framework of Chen and Tell, using a variant of the Shaltiel--Umans generator

The Cerenkov effect revisited: from swimming ducks to zero modes in gravitational analogs

We present an interdisciplinary review of the generalized Cerenkov emission of radiation from uniformly moving sources in the different contexts of classical electromagnetism, superfluid hydrodynamics, and classical hydrodynamics. The details of each specific physical systems enter our theory via the dispersion law of the excitations. A geometrical recipe to obtain the emission patterns in both real and wavevector space from the geometrical shape of the dispersion law is discussed and applied to a number of cases of current experimental interest. Some consequences of these emission processes onto the stability of condensed-matter analogs of gravitational systems are finally illustrated.Comment: Lecture Notes at the IX SIGRAV School on "Analogue Gravity" in Como, Italy from May 16th-21th, 201

Investigating the Gamma-ray Strength Function in 74Ge using the Ratio Method

>Magister Scientiae - MScAn increasing number of measurements reveal the presence of a low-energy enhancement in the gamma-ray strength function (GSF). The GSF, which is the ability of nuclei to absorb or emit rays, provides insight into the statistical properties of atomic nuclei. For this project the GSF was studied for 74Ge which was populated in the reaction 74Ge(p,p')74Ge* at a beam energy of 18 MeV. The data were collected with the STARS-LIBERACE array at Lawrence Berkeley National Laboratory. Silicon detector telescopes were used for particle identi cation and rays in coincidence were detected with 5 clover-type high-purity germanium detectors. Through the analysis particle- - coincidence events were constructed. These events, together with well-known energy levels, were used to identify primary rays from the quasicontinuum. Primary rays from a broad excitation energy region, which decay to six 2+ states could be identi ed. These states and the associated primary rays are used to measure the GSF for 74Ge with the Ratio Method [1], which entails taking ratios of e ciency-corrected primary -ray intensities from the quasicontinuum. Results from the analysis of the data and focus on the existence of the low-energy enhancement in 74Ge will be discussed. The results are further discussed in the context of other work done on 74Ge using the ( , ') [2], (3He,3He') [3] and ( , ') [4] reactions

Modelling FX smile : from stochastic volatility to skewness

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